Strictly speaking they require /a/ Social Security Number, not necessarily /your/ SSN.
When I signed up for my most recent cellular service, I hesitated at giving my information to the guy in the store. I told him that since it was a pre-paid account I wasn't asking for credit, so there's no need for him to have my information.
He was OK with that, and pulled out an ID card the store keeps in a drawer for just such occasions.
Salespeople have a sales quota, not an enforce-the-rules quota.